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Weak Error for stable driven SDEs: expansion of the
densities
Valentin Konakov, Stephane Menozzi
To cite this version:
Valentin Konakov, Stephane Menozzi. Weak Error for stable driven SDEs: expansion of the densities.
Journal of Theoretical Probability, Springer, 2011, 24 (2), pp.454-478. �hal-00331845v3�
densities.
Valentin Konakov · St´ephane Menozzi
January 22, 2010
Abstract Consider a multidimensional SDE of the form Xt = x +R0tb(Xs−)ds +
Rt
0f (Xs−)dZs where (Zs)s≥0 is a symmetric stable process. Under suitable
assump-tions on the coefficients the unique strong solution of the above equation admits a density w.r.t. the Lebesgue measure and so does its Euler scheme. Using a parametrix approach, we derive an error expansion w.r.t. the time step for the difference of these densities.
Keywords Symmetric stable processes · parametrix · Euler scheme Mathematics Subject Classification (2000) 60H30 · 65C30 · 60G52
1 Introduction
Consider the followingRd-valued Stochastic Differential Equation (SDE in short)
Xt= x + Z t 0 b(Xs−)ds + Z t 0 f (Xs−)dZs, (1.1)
where b, f are respectively Lipschitz continuous mappings from Rd toRd and Rd to Rd⊗Rd and (Zs)s≥0 is a general L´evy process. The previous assumptions guarantee
the existence of a unique strong solution to (1.1). Also, this solution satisfies the strong Markov property, see e.g. Theorem 7 and 32 Chapter 5 in Protter [Pro04]. Let T > 0 be a fixed time horizon and (XtN)t∈Λ a given approximation scheme of (Xt)t∈[0,T ]
associated to the time step h := T /N, N ∈ N∗ on the grid Λ := {ti := ih, i ∈
[[0, N ]]}. When speaking about weak approximation of (1.1) two kinds of quantities are of interest. The first one writes
E1(x, T, N ) :=Ex[g(XT)] −Ex[g(XTN)]
V. Konakov,
CEMI, Academy of Sciences. Nahimovskii av. 47, 117418 Moscow, RUSSIA. E-mail: valentin [email protected]
S. Menozzi,
LPMA, Universit´e Paris VII Diderot. 175 Rue du Chevaleret, 75013 Paris, FRANCE. E-mail: [email protected]
for a suitable class of test functions g. The second one concerns, when it exists, the approximation of the transition density p of the original SDE (1.1). If the approximation scheme (XtN)t∈Λ admits as well a transition density pN, the quantity under study
becomes
E2(x, y, T, N ) := (p − pN)(T, x, y).
In both cases, the goal is to give a bound or an error expansion of these quantities in terms of h. The error expansions are particularly useful for practical simulation. For E1, the expansion allows to use the Romberg Richardson extrapolation to improve
the convergence of the discretization error, see e.g. Talay and Tubaro [TT90]. On the other hand, if p and pN exist, and a suitable expansion of E2holds, it can be useful to
estimate the sensitivity of E1w.r.t. to the spatial variable x and it also allows to get a
control on E1for a wider class of test functions g than those considered by the direct
methods used to control this quantity, see e.g. Guyon [Guy06]. Indeed, the typical assumptions and techniques associated to the study of E1and E2are quite of different
nature.
In the continuous case, i.e. Zs = bs + σWs where (Ws)s≥0 is a standard
d-dimensional Brownian motion, provided the test function g and the coefficients b, f are sufficiently smooth and g has polynomial growth, without any additional assump-tion on the generator Talay and Tubaro [TT90] derive an error expansion at order 1 for E1(x, T, N ) when (XtN)t∈Λ is the Euler approximation. Their proof is based on
standard stochastic analysis tools: Itˆo’s expansions and stochastic flows. To obtain the same kind of result for bounded Borel functions g some non degeneracy has to be assumed, namely hypoellipticity of the underlying diffusion, and the proof relies on Malliavin calculus techniques, see Bally and Talay [BT96a]. The authors also manage to extend their results to E2(x, y, T, N ) for a slightly modified Euler scheme [BT96b].
Anyhow, in the uniformly elliptic case, the most natural approach to handle the estimation of the quantity E2(x, y, T, N ) consists in using the so called ”parametrix”
technique introduced to obtain existence and controls on the fundamental solutions of PDEs, see e.g. Mc Kean and Singer [MS67] or Friedman [Fri64]. Roughly speaking it consists in expressing the density of XT in terms of an infinite sum of suitable iterated
kernels applied to the density of an SDE with constant coefficients. This has been done successfully by Konakov and Mammen [KM02].The main advantage of this approach is that the density of the solution XT and the Euler approximation XTN can be expressed
in the same form and therefore quite directly compared. Furthermore this technique turns out to be quite robust and can be applied as soon as good controls on the densities p, pN and their derivatives are available, see e.g. [KMM08] for an extension to a slightly degenerate framework.
For a general L´evy process Z and suitable smooth functions b, f , g, under additional assumptions on the behavior at infinity of the L´evy measure ν of Z, that is integrability conditions of the large jumps, Protter and Talay [PT97], manage to get a control at order one or even an error expansion for E1(x, T, N ) with the same approach as in
[TT90]. In that work the approximation is the Euler scheme which for a general L´evy measure ν cannot always be exactly simulated on a computer.
The quantity E1(x, T, N ) for approximations of the Euler scheme that can be
sim-ulated has also been studied by Jacod et al. [JKMP05] who derived bounds at order 1. Moment conditions are also assumed. We finally refer to the work of Hausenblas and Marchis [HM06] for approximations of Poisson jump measures that are easy to simulate.
In this work, we consider the case where (Zt)t≥0is an α-stable symmetric process,
α ∈ (0, 2). Under suitable non degeneracy assumptions on its coefficients specified below (see (A-1)-(A-3)), equation (1.1) is known to have a density p w.r.t. the Lebesgue measure. This can be proved via a Malliavin calculus-Bismut integration by parts approach, see e.g. Bichteller et al. [BGJ87]. Also, a direct construction of this density using a parametrix expansion has been obtained by Kolokoltsov [Kol00] who derived as well ”Aronson’s like” bounds with time singularity depending on the index α of the stable process (Zt)t≥0.
Analogoulsy to the ”diffusion case” the first step of the parametrix is to consider that the density p(T, x, y) of (1.1) can be approximated by the density of the process
e
Xty = x + b(y)t + f (y)Ztat time T . Namely, we freeze the coefficients in (1.1) at the
final spatial point. The next crucial point is to obtain sharp estimates of the stable densitypey(T, x, .) of XeTy and its derivatives in order to solve the parametrix integral equations.
Stable driven SDEs appear in various applicative fields, from mathematical physics to electrical engineering or financial mathematics, see [IP06], [SK74] or [JMW96], there-fore their approximation becomes of interest. To approximate equation (1.1), setting φ(t) := inf{ti: ti≤ t < ti+1}, we introduce the Euler scheme
XtN = x + Z t 0 b(Xφ(s)N )ds + Z t 0 f (Xφ(s)N )dZs. (1.2)
The computation of the above scheme only requires to be able to simulate exactly the increments of (Zt)t≥0, which up to a self similarity argument only amounts to simulate
a stable law. This aspect is for instance discussed in Samorodnitsky and Taqqu [ST94], Weron and Weron [WR95] or Section 3 of [PT97]. Under the same assumptions (A-1)-(A-3), the Euler scheme defined in (1.2) also has a density pN.
Observe that the results of [PT97], [JKMP05] cannot be directly applied, even for the study of E1(x, T, N ), since stable laws have heavy tails. Comparing the parametrix
developments of p and pN we obtain an expansion with leading term of order 1 in h for E2(x, y, T, N ). The parametrix expansion of p is discussed in [Kol00], see also Section
3 and Appendix, whereas the parametrix expansion of pN can be related to the ideas developed in [KM00, KM02] for the diffusive case corresponding to an index of stability equal to 2.
This result also emphasizes the robustness of the method that naturally extends to a broad class of processes. Let us mention that, using a Malliavin calculus approach, Hausenblas [Hau02], derived an upper bound of order one w.r.t. h for the quantity E1(x, T, N ), g ∈ L∞ in the scalar case. Concerning functional limit theorems for the
approximation of stable driven SDEs we refer to the work of Jacod [Jac04].
The paper is organized as follows. In Section 2 we state our standing assumptions and main results. In Section 3 we prove the existence of the densities for both the stable driven equation and its Euler scheme and also give a parametrix representation of these densities. Section 4 is dedicated to the proof of the main results. Eventually, we state in Section 5 weaker assumptions under which our main result holds and we also briefly discuss how to extend it to the case of a stable process perturbed by a compound Poisson process.
2 Assumptions and Main results 2.1 Assumptions and Notation
In the following we consider symmetric stable processes, that is, for all t ≥ 0, u ∈Rd, E[exp(ihu, Zti)] = exp(ithγ, ui + t
Z
Sd−1
Z +∞ 0
eiρhu,si− 1 − ihu, ρsi 1 + ρ2 dρ ρ1+αλ(ds))˜ = exp(ithγ, ui − t Z Sd−1 |hs, ui|αλ(ds)), (2.1) where ˜λ is a symmetric measure on the unit sphere Sd−1 (i.e. for every A in the Borel σ-field B(Sd−1), ˜λ(A) = ˜λ(−A)). The second equality in equation (2.1) is then obtained by direct integration over ρ and λ = Cα˜λ with
Cα:= Γ (1 − α)α−1cos πα 2 Iα6=1+π 2Iα=1.
We refer to the proof of Theorem 9.32 in Breiman [Bre68] and Lemma 2, Chapter XVII.4 in Feller [Fel66] for the expression of Cα.
We now introduce our assumptions. Fix an integer q ≥ 2. We assume that (A-1) For d ≥ 2, the spherical measure λ has a Cq(Sd−1) surface density and for all d ≥ 1, there exist constants 0 < C1≤ C2< +∞, ∀p ∈Rd,
C1|p|α≤
Z
Sd−1
|hp, si|αλ (ds) ≤ C2|p|α.
(A-2) The coefficients b and f and their derivatives up to order q are uniformly bounded in x. Thus, for 1 < α < 2, B(x) := b(x) + f (x)γ is uniformly bounded. We impose for 0 < α ≤ 1, B(x) = 0 for all x ∈Rd.
(A-3) There exist constants 0 < c ≤ c < +∞ s.t. for all x ∈Rd, ξ ∈Rd,
c|ξ|2≤ hf (x)ξ, ξi ≤ c|ξ|2.
From now on we assume that Assumptions (A-1)-(A-3) are in force.
Remark 21 Note that for d = 1, with the convention S0= {−1, 1}, we have C1= C2
in (A-1) even without symmetry. The symmetry is actually not needed in that case, see the beginning of Section 3 in [Kol00].
Remark 22 The zero drift condition in (A-2) comes from the fact that for α ∈ (0, 1] the addition of a drift of order t does not correspond to a negligible term in small time with respect to the natural scale t1/α, see Appendix B in [KM10] for details.
In the following we denote by C a positive generic constant that can depend on α, d, the bounds appearing in the previous assumptions but neither on N nor on the spatial points involved. Its value may change from line to line. Other possible dependencies, especially w.r.t. the final time T are explicitely specified. Concerning functional spaces, we denote by Cbk(Rd), k ∈ N∗, the Banach space of continuous bounded functions
having bounded derivatives up to and including the order k with the norm kf k := max0≤l≤ksupx∈Rd|f(l)(x)|. Eventually C0k(Rd) stands for the functions in Cbk(Rd)
2.2 Generator
From equation (2.1) and standard computations, see e.g. equation (5.11) in [JKMP05], we derive that for every smooth function g ∈ C02(Rd), the generator of (1.1) writes
Φg(x) = hB(x), ∇xg(x)i −
Z
Rd
g(x + f (x)y) − g(x) −h∇xg(x), f (x)yi 1 + |y|2 ν(dy),
where B(x) = b(x) + f (x)γ and ν stands for the L´evy measure of Z. Introduce for all A ∈ B(Rd), νf (x)(A) := ν({y ∈Rd : f (x)y ∈ A}) and denote by ˜λf (x) its spherical
part (which is still a symmetric measure). Setting z = f (x)y in the above equation, using the symmetry and the polar coordinates we derive:
Φg(x) = hB(x), ∇xg(x)i + Z Sd−1 Z +∞ 0 g(x + ρs) − g(x) −ρh∇xg(x), si 1 + ρ2 dρ ρ1+αλ˜f (x)(ds). (2.2)
Remark 23 Denote similarly to (2.1), λf (x)= Cα˜λf (x). The uniform ellipticity
con-dition (A-3) allows to have good controls on the measure λf (x)(·). As a consequence
of (A-1), (A-3) one gets that there exist constants 0 < C1 = C1(c, d, α) ≤ C2 =
C2(c, d, α) < +∞ s.t. ∀p ∈Rd, x ∈Rd, C1|p|α≤ Z Sd−1 |hp, si|αλf (x)(ds) ≤ C2|p|α. (2.3) 2.3 Main results
Proposition 21 For every t > 0 the solution Xt (resp. XtN) of (1.1) (resp. (1.2))
has a density p(t, x, ·)(resp. pN(t, x, ·)) w.r.t. the Lebesgue measure. Additionally, as a function of the space variables the density p is in Cbq(Rd×Rd) if α > 1 and in
Cbq−1(Rd×Rd) if α ≤ 1.
To state the theorem we first need some notation. Introduce for all ξ ∈ Rd and all
smooth function ϕ(t, x, y) the integro-differential operators:
e Φξϕ(t, x, y) = hB(ξ), ∇xϕ(t, x, y)i + Z Sd−1 Z +∞ 0 g(x + ρs) − g(x) − ρh∇xg(x), si 1 + ρ2 dρ ρ1+α˜λf (ξ)(ds). (2.4)
With this definition we write for given (x, y) ∈Rd:
e Φ∗ϕ(t, x, y) =Φeyϕ(t, x, y), ∀m ∈N∗, e Φ∗ m ϕ(t, x, y) =Φeξ m ϕ(t, x, y) |ξ=x, (2.5) Note that we haveΦe∗ϕ(t, x, y) = Φϕ(t, x, y) defined in (2.2) but in general, for m ≥ 2,
e
Φ∗
m
ϕ(t, x, y) 6= (Φ)mϕ(t, x, y). Define now, for t > 0, the kernel
whereepy(t, x, y) denotes the density at point y ofXet= x+b(y)t+f (y)Zt. Note that the
variable y acts here twice: as the argument of the density and as a defining quantity of the processXet(≡Xet,x,y), i.e. the coefficients are frozen in y. Eventually we introduce
the continuous and discrete convolution operators
ϕ ⊗ ψ(t, x, y) = Z t 0 du Z dzϕ(u, x, z)ψ(t − u, z, y), ∀t ∈ [0, T ], ϕ ⊗N ψ(t, x, y) = Z t 0 du Z
dzϕ(φ(u), x, z)ψ(t − φ(u), z, y), ∀t ∈ {(ti)i∈[[1,N ]]},
with φ(u) is defined just before (1.2) and denotes the largest discretization time lower or equal to u. Also ϕ ⊗ H(0) = ϕ and ϕ ⊗ H(r) =ϕ ⊗ H(r−1)⊗ H stands for the r-fold convolution.
Theorem 21 Suppose q > d + 4. Take 0 < M ≤ q − (d + 4). There exists a function RM(T, x, y) with |RM(T, x, y)| ≤ CM(T ) 1 1+|y−x|d+α := ρα,M(T, y − x) for some
positive constant CM(T ) such that
(p − pN)(T, x, y) = M−1X l=1 hl (l + 1)! p ⊗N Φ −Φe∗l+1pd (T, x, y) − − M−1X k=1 hk (k + 1)! pd⊗N e Φ∗−Φe∗ k+1 pN (T, x, y) + hMRM(T, x, y)
with P0l=1· · · = 0 and ∀t ∈ {(ti)i∈[[1,N ]]}, pd(t, x, y) :=P∞r=0
e p ⊗NH(r) (t, x, y). It holds that M−1X l=1 p ⊗N Φ −Φe∗l+1pd (T, x, y) ≤ ρα,M(T, y − x), M−1X k=1 pd⊗N e Φ∗−Φe∗ k+1 pN (T, x, y) ≤ ρα,M(T, y − x).
Remark 24 In the above expression, one writes for all l ∈ [[1, M − 1]],
Φ −Φe∗l+1ϕ(t, x, y) = l+1 X k=1 Cl+1k Φk(−Φe∗)l+1−kϕ(t, x, y), whereas, ∀k ∈ [[1, M − 1]], e Φ∗−Φe∗ k+1 ϕ(t, x, y) =(Φeξ−Φey) · · · (Φeξ−Φey) | {z } (k+1) times ϕ(t, x, y)|ξ=x = (Φeξ−Φey)k+1ϕ(t, x, y) ξ=x.
Remark 25 The terms in the previous expansion depend on N . Anyhow using itera-tively the Theorem and controls on ⊗N− ⊗ (see also Lemma 41) it is possible to obtain
an expansion with terms independent of N . For small M explicit formulas are thus easily derived but in all generality the terms become less transparent. For M = 2 one gets (p − pN)(T, x, y) =h 2 p ⊗N(Φ −Φe∗)2pd− pd⊗N (Φe∗−Φe∗)2pN (T, x, y) +h2R2(T, x, y) =h 2 p ⊗ (Φ −Φe∗)2p − p ⊗ (Φe∗−Φe∗)2p (T, x, y) +h2Re2(T, x, y) =h 2(p ⊗ (Φ 2 −Φe2∗)p)(T, x, y) + h2Re2(T, x, y),
whereRe2(T, x, y) ≤ C(T )ρα,2(T, y − x) for some positive constant C(T ).
From the above expansion and the controls on the density and its derivatives, see e.g. Theorems 3.1, 3.2 and Proposition 3.1 in [Kol00] or Lemma 43, we can derive the error expansion for E1(x, T, N ) for measurable functions g satisfying the growth
condition ∃C > 0, |g(x)| ≤ C(1 + |x|β), β < α. In particular, we do not need the smoothness assumption on g required in the approach of [TT90], [PT97]. We recall that the expansion of E1(x, T, N ) allows from a practical point of view to improve the
convergence rate of the discretization error using the Romberg Richardson extrapolation. This simply consists in observing that the expansion yieldsE[g(XT)] − (E[2g(X2N
T )] −
E[g(XN
T )]) = O(h2). The associated Monte Carlo estimator, involving a refined scheme,
is then used for simulations see [TT90] for details.
Also, the expansion can be used to study the sensitivity of E1(T, x, N ) w.r.t. x
with-out any additional assumption on g. This is crucial for financial applications (hedging), see e.g. Guyon [Guy06] for further developments in the diffusive case.
3 Stable driven equations and their Euler scheme: existence of the density and associated parametrix expansion
3.1 Stable driven equation
3.1.1 Proof of Proposition 21: existence of the density for the solution of (1.1)
For (Xt)t≥0, the existence of the density derives from Proposition 3.4 in [Kol00], where
some properties of the fundamental solution of ∂tp(t, x, y) = Φp(t, x, y), p(0, x, y) =
δ(y −x) are discussed, and a standard identification argument, see e.g. Dynkin [Dyn63], Theorem 2.3, p. 56. The stated smoothness of the density is then a consequence of point (ii) of the same Proposition.
Remark 31 The existence of the density is discussed in Bichteler et al. [BGJ87], where it is proved thanks to a Bismut-Malliavin approach. This technique requires the computation of a tangent equation associated to the gradient flow that involves the derivatives of the coefficients of equation (1.1). Thus, some additional smoothness of the coefficients is needed, see e.g. Theorem 6.48 of the above reference. We also mention the result of Picard [Pic96], Theorem 4.1, that gives existence and smoothness of the density for L´evy driven SDEs for very singular L´evy measures, provided there are sufficiently small jumps. For smooth coefficients b, f , it includes in particular the case of (1.1) where the spherical measure λ can be atomic.
3.1.2 Parametrix expansion of the density
For the sake of completeness and also because it is crucial for the discrete model we briefly recall how to get through a ”parametrix” approach a series expansion for the density p(t, x, y).
Introduce, for all x, y ∈Rdthe following stochastic ”frozen” stable driven equation e
Xt≡Xet,x,y defined for t ≥ 0 by
e Xt= x + Z t 0 b(y)du + Z t 0 f (y) dZu. (3.1)
By computation of the Fourier transform of Zt and Fourier inversion the transition
densityepy(t, x, z) ofXetat point z ∈Rd explicitly writes
e py(t, x, z) = 1 (2π)d Z e−ihz−x−tB(y),piexp −t Z Sd−1 |hp, si|αλf (y)(ds) dp, (3.2) where λf (y)has been introduced in Section 2.2. The densities of the solutions of (3.1)
and (1.1) satisfy respectively ∂epy ∂t (t, x, z) =Φeype y (t, x, z), for t > 0, (x, z) ∈ (Rd)2, pey(0, x, z) = δ(z − x), ∂p ∂t(t, x, z) = Φp(t, x, z), for t > 0, (x, z) ∈ (R d )2, p(0, x, z) = δ(z − x). (3.3) Note carefully that the derivatives inΦey are taken w.r.t. the x variable.
We will speak about the operators appearing in (3.3) as the ”frozen” and ”unfrozen” ones. In the following ∀(t, x, z) ∈R+∗×(Rd)2,p(t, x, z) :=e pez(t, x, z). Hence, from (2.6)
∀(t, z, y) ∈R+∗× (Rd)2, H(t, z, y) = (Φ −Φey)p(t, z, y) = (e Φez−Φey)p(t, z, y).e
Proposition 31 (Parametrix expansion of the density) With the notations of Sec-tion ??, the following representaSec-tion holds
p(t, x, y) =
∞
X
r=0
(p ⊗ He (r)) (t, x, y) . (3.4)
Proof. Equations (3.3) correspond to the forward Kolmogorov equations. Consider now the backward equation for p, namely, ∂sp(s, x, z) =tΦezp(s, x, z) wheretΦez stands for
the adjoint operator ofΦezand the derivatives are taken w.r.t. z. Differentiating under
the integral we have from (3.3)
(p −ep) (t, x, y) = Z t 0 ds∂ ∂s Z p (s, x, z)ep (t − s, z, y) dz = Z t 0 ds Z [tΦezp (s, x, z)p (t − s, z, y) − p (s, x, z)e Φeyp (t − s, z, y)]dz = p ⊗ H(t, x, y).e
The representation (3.4) then follows by simple iteration.
Remark 32 Note that the previous expansion is ”formal”. The convergence of the r.h.s. in (3.4) is investigated in the proof of Theorem 3.1 in [Kol00] and can also be derived with the controls of Lemmas A1 and A2 below. For the sake of completeness, a short proof of this convergence is also given in Appendix B.
3.2 Euler scheme
We consider now, for given N ∈N∗, the Euler scheme for equation (1.1) at the
dis-cretization times: X0N = x, XtNi+1 = X N ti + b(X N ti)h + f XtNi Zti+1− Zti recalling h = T /N .
3.2.1 Proof of Proposition 21 for the Euler scheme: existence of the density
For each N ∈ N∗, (XtN
i)i∈[[0,N ]] is a Markov chain. Given the past {X
N
tl = xl, l ∈
[[0, i]]}, the conditional distribution of the innovations b(XtNi)h + f
XtNi
Zti+1− Zti
has conditional densityepxi(h, 0, ·) (with the notation of (3.1), (3.2)). This proves the
existence of the density for the discretization scheme. 3.2.2 Parametrix expansion for the Euler scheme
To give for the Euler scheme an expansion similar to equation (3.4), that will also be the starting point for our error expansion, we need to define, for fixed j, k, 0 ≤ j < k ≤ N and x, y ∈Rdadditional ”frozen” Markov chains (XetN
l)l∈[[j,k]]= (Xe N tl,x,y)l∈[[j,k]]. Their dynamics is described by e XtNj = x,Xe N ti+1=Xe N
ti + b(y)h + f (y) Zti+1− Zti
, i ∈ [[j, k − 1]].
Given the past {XetNl = xl, l ∈ [[j, i]]}, the conditional distribution of the innovations
b(y)h+f (y)(Zti+1−Zti) has conditional densitype
y(h, 0, ·) and, hence, does not depend
on the past. Note that for the grid points (ti)i∈[[0,N ]] the transition densities of the
solution Xes,x,y of (3.1) coincide with the transition densities of the chainXetNj,x,y for
N ∈N∗, x, y ∈Rdand s = tj.
For all 0 ≤ j < k ≤ N, (x, y) ∈ (Rd)2, we denote by pN tk− tj, x, y and e
pN(tk− tj, x, y) the transition probability densities between times tjand tkfrom point
x to y of the chains XN andXeN respectively. In particular,
e
pN(tk− tj, x, y) =pey(tk− tj, x, y) =p(te k− tj, x, y). (3.5)
Before stating the parametrix expansion of pN in terms ofepN, we need to introduce a kernel HN that is the ”discrete” analogue of H defined in (2.6):
HN(tk− tj, x, y) = n LN−LeyN o e pN(tk− tj, x, y), (3.6) with LNϕ(tk− tj, x, y) = h−1{ Z pN(h, x, z)ϕ(tk− tj+1, z, y)dz − ϕ(tk− tj+1, x, y)}, e LyNϕ(tk− tj, x, y) = h−1{ Z e py(h, x, z)ϕ(tk− tj+1, z, y)dz − ϕ(tk− tj+1, x, y)}.
Note that the previous definitions yield pN(h, x, z) =epx(h, x, z). We also mention that, because of the discretisation, there is a slight ”shift” in time in the definition of HN.
Lemma 31 For 0 ≤ j < k ≤ N the following formula holds: pN(tk− tj, x, y) = k−j X r=0 (peN⊗NHN(r))(tk− tj, x, y) (3.7)
where in the calculation ofpeN ⊗N HN(r) (r-fold convolution) we define
pN(0, x, y) =epN(0, x, y) = δ(x − y).
The proof of this lemma is given in [KM00], Lemma 3.6 and does not rely on the specific distribution of the innovations.
Remark 33 With the convention that HN(r) = 0 for r > k − j, equation (3.7) also writes pN(tk− tj, x, y) =P∞r=0(peN⊗N HN(r))(tk− tj, x, y). This expression will often
be used in the sequel.
4 Proof of the main results
In this section, we state in Subsection 4.1 the various points needed to prove Theorem 21. The proofs are postponed to Subsection 4.2. As mentioned earlier, the key idea consists in comparing the parametrix expansions of the densities p and pN respectively given by (3.4) and (3.7). In the whole section we suppose that the assumptions of Theorem 21 hold.
4.1 Proof of Theorem 21
For the previously mentioned comparison to be possible we first need to estimate a difference between the transition density p(T, x, y) and pd(T, x, y) := X
r≥0
e
p ⊗N
H(r)(T, x, y) which is the analogous of (3.4) up to the discrete time convolution (i.e. ⊗ replaced by ⊗N). We refer to (2.2), (2.4), (2.5), (2.6) for the definition of operators
and kernels.
Lemma 41 (Time discretization) One has:
(p − pd)(T, x, y) = M−1X l=1 hl (l + 1)! p ⊗N Φ −Φe∗l+1pd (T, x, y) +hMRM,1(T, x, y) with M−1X l=1 p ⊗N Φ −Φe∗l+1pd (T, x, y) + |RM,1(T, x, y)| ≤ ρα,M(T, y − x).
Lemma 42 (Comparison of the discrete convolutions) The following expansion holds: (pd− pN)(T, x, y) = − M−1X k=1 hk (k + 1)! pd⊗N e Φ∗−Φe∗ k+1 pN (T, x, y) +hMRM,2(T, x, y) where RM,2(T, x, y) = − 1 M ! Z 1 0 (1 − τ )M pd⊗N e Φ∗−Φe∗ M+1 e p∆τ (T, x, y)dτ, ∀t ∈ {(ti)i∈[[1,N ]]}, pe∆τ (ti, x, y) = ∞ X r=0 e pτ⊗NHN(r)(ti, x, y) ,pe∆0 = pN, and ∀τ ∈ [0, 1], peτ(t, x, y) = Z Rde px(τ h, x, z)pey(t − τ h, z, y)dz.
In particularpe0(t, x, y) =pey(t, x, y). Also, M−1X k=1 pd⊗N e Φ∗−Φe∗ k+1 pN (T, x, y) + |RM,2(T, x, y)| ≤ ρα,M(T, y − x).
Theorem 21 is then a direct consequence of Lemmas 41 and 42.
4.2 Proofs of the technical Lemmas
Proof of Lemma 41. We start from the recurrence relation for r ∈N∗ e p ⊗ H(r)−p ⊗e NH(r)= h e p ⊗ H(r−1)⊗ H −ep ⊗ H(r−1)⊗NH i +hp ⊗ He (r−1)−p ⊗e N H(r−1) i ⊗N H.
Summing up these terms over r ∈ N∗ and using the linearity of ⊗ and ⊗N we get
p − pd= p ⊗ H − p ⊗NH +
p − pd⊗N H. An iterative application of this identity
yields (p − pd)(T, x, y) = ∞ X r=0 [p ⊗ H − p ⊗NH] ⊗N H(r)(T, x, y). (4.1)
By definition, for all k ∈ [[1, N ]],
[p ⊗ H − p ⊗NH](tk, x, y) = k−1 X j=0 Z tj+1 tj ds Z [p (s, x, z) H (tk− s, z, y) −p tj, x, zH tk− tj, z, y]dz. (4.2)
A Taylor expansion of the function θ(s, z) := p(s, x, z)H(tk− s, z, y) in the interval [tj, s] ⊆ [tj, tj+1] gives Z [θ(s, z) − θ(tj, z)]dz = M−1X l=1 (s − tj)l l! Z ∂τlθ(τ, z) τ =tj dz + (s − tj)M (M − 1)! Z 1 0 (1 − δ)M−1 Z ∂τMθ(τ, z)|τ =τj(s,δ)dzdδ, (4.3)
where τj(s, δ) = tj+ δ(s − tj). Note now that −∂sp(t − s, x, z) = Φp(t − s, x, z), ∂tp(t −
s, x, z) =tΦp(t − s, x, z). HeretΦ =tΦezis the adjoint operator of Φ where the
deriva-tives have to be taken w.r.t. z. Hence, Φp(t−s, x, z) =tΦp(t−s, x, z). The same identity also holds forp with Φ,e tΦ respectively replaced byΦe∗,tΦe∗. We therefore derive
Z ∂τθ(τ, z)|τ =tjdz = Z ∂τ[p(τ, x, z)] |τ =tjH(tk− tj, z, y)dz + Z p(tj, x, z)∂τ[H(tk− τ, z, y)] |τ =tjdz = Z t Φzp(tj, x, z) Φ −Φe∗ep(tk− tj, z, y)dz − Z p(tj, x, z) Φ −Φe∗Φe∗p(te k− tj, z, y)dz = Z p(tj, x, z) Φ −Φe∗2p te k− tj, z, ydz.
Iterating the differentiation we get
Z ∂lτθ(τ, z)|τ =tjdz = Z p(tj, x, z) Φ −Φe∗ l+1 e p(tk− tj, z, y)dz, (4.4)
where we recall that for two operators A and B we denote by (A − B)k the following sum (A − B)k =Pkj=0CkjAk−j(−B)j.
Plugging (4.3) and (4.4) into (4.2) we get [p ⊗ H − p ⊗N H](tk, x, y) = M−1X l=1 hl (l + 1)!p ⊗N Φ −Φe∗l+1p (te k, x, y) +hMReM,1(tk, x, y) (4.5) where e RM,1(tk, x, y) = 1 (M − 1)! k−1X j=0 Z tj+1 tj h h−1(s − tj) iMZ 1 0 (1 − δ)M−1× Z ∂τM[p (τ, x, z) H (tk− τ, z, y)] |τ =τj(s,δ)dsdzdδ. (4.6)
Plugging (4.5) and (4.6) into (4.1) we get (p − pd)(T, x, y) = M−1X l=1 hl (l + 1)!× ∞ X r=0 p ⊗N Φ −Φe∗ l+1 e p ⊗NH(r)(T, x, y) +hMRM,1(T, x, y) (4.7)
with RM,1(T, x, y) =P∞r=0(ReM,1⊗NH(r))(T, x, y).
Now we apply that for a linear operator S and its adjointtS we have p ⊗NSp =e tSp ⊗ N ep. This gives ∞ X r=0 p ⊗N Φ −Φe∗ l+1 e p ⊗N H(r)(T, x, y) = t Φ −Φe∗l+1 p ⊗N ∞ X r=0 e p ⊗NH(r) (T, x, y) = p ⊗N Φ −Φe∗l+1pd(T, x, y),
which plugged into (4.7) gives the desired expansion. The stated bound follows by application of the estimates given in Lemma 43 below. We only give the proof for the first summand, the other terms of the sum over l and the remainder RM,1(T, x, y) can
be handled in a similar way. Write
p ⊗N (Φ −Φe∗)2pd(T, x, y) = N −1X j=0 h Z p(tj, x, z)(Φ −Φe∗)2pd(T − tj, z, y)dz := S1+ S2,
where in S1 (resp. S2) the sum is taken over I1 := {j ∈ [[0, ⌊N −12 ⌋]]} (resp. I2 :=
{j ∈ [[⌊N −12 ⌋ + 1, N − 1]]}). For S1(resp. S2), pd(T − tj, z, y) (resp. p(tj, x, z)) is non
singular. From Lemma 43 below equation (4.12), there exists C := C(T ) s.t. for all (x, y, z) ∈ (Rd)3, p(s, x, z) ≤ Cepy(s, x, z), pd(s, z, y) ≤ Cepy(s, z, y), ∀s ∈]0, T ], |(Φ −Φe∗)2pd(T − tj, z, y)| ≤ Cepy(T − tj, z, y), ∀j ∈ I1, t Φ −Φe∗2 p(tj, x, z) ≤ Cpey(tj, x, z), ∀j ∈ I2. (4.8)
The semigroup property for ˜py yields |S1| + |S2| ≤ Cp(T, x, y). One eventually checkse
from Proposition B1 thatp(T, x, y) :=e pey(T, x, y) ≤ ρα,M(T, y − x).
Proof of Lemma 42. Let us denote by F[ψ](z) =Rexp(ihz, pi)ψ(p)dp the Fourier transform of a function ψ. Introduce now for all u, t, u < t, u, t ∈ {(ti)i∈[[0,N ]]}, p ∈Rd,
ψ(p) = h(LN −LeyN)ep
y(t − u, x, p)
=
Z
pN(h, x, w)pey(t − (u + h), w, p)dw −pey(t − u, x, p).
Note that in particular according to (3.6), ψ(y) = hHN(t − u, x, y). Taking the
charac-teristic functions of the densities involved in the above equation, we obtain from (3.2) and (3.6) that
F[ψ](z) := Gz(1) − Gz(0)
with
Gz(τ ) = exp
ihx, zi + i(t − u)hB(y), zi + iτ hh∆Bx,y, zi −
Z
Sd−1
|hz, si|α(t − u)λf (y)(ds) + τ h∆λx,y(ds)
where ∆Bx,y = B(x) − B(y), ∆λx,y(ds) = λf (x)(ds) − λf (y)(ds). Note in particular that ∀τ ∈ [0, 1], Gz(τ ) = Gz(0) × exp τ h ih∆Bx,y, zi − Z Sd−1 |hz, si|α∆λx,y(ds) . (4.9) A Taylor expansion yields F[ψ](z) =PMk=1k!1G(k)z (0) +M!1
R1 0(1 − τ )
MG(M+1) z (τ )dτ .
From (4.9), one derives that for k ∈N∗:
1 k!G (k) z (0) = h k k!Gz(0) i∆Bx,y, z − Z Sd−1 |hz, si|α∆λx,y(ds) k .
Observe now that Gz(0) = F[θ](z), θ(p) := epy(t − u, x, p). Using the well-known
properties of the Fourier transform one gets for all k ∈ [[1, M ]]
G(k)z (0) = hkF e Φξ−Φey k θ ξ=x (z),
where the operatorsΦe.are applied w.r.t. the x component and the Fourier transform is
applied w.r.t. the p component ofpey(t−u, x, p). Also, in the above writing, we compute the Fourier transform for an arbitrary fixed ξ ∈Rdand we then put ξ = x.
Hence, F[ψ](z) = PMk=1k!1G(k)z (0) +M!1 R01(1 − τ )MG(M+1)z (τ )dτ = PM k=1h k k!F e Φξ−Φey k θ ξ=x (z) + hM +1 M! F " R1 0(1 − τ ) M Φe ξ−Φey M+1 θτ ξ=x dτ # (z), where ∀τ ∈ [0, 1], θτ(p) := Z Rde
px(τ h, x, z)pey(t − u − τ h, z, p)dz. Taking the inverse Fourier transform and putting p = y in the above equation, observing that H(t − u, x, y) = (Φe∗−Φe∗)pey(t − u, x, y), we obtain (HN− H)(t − u, x, y) = M−1X k=1 hk (k + 1)! e Φ∗−Φe∗ k+1 e p(t − u, x, y) + hM M ! Z 1 0 (1 − τ )MΦe∗−Φe∗ M+1 e pτ(t − u, x, y)dτ . (4.10)
Recall now that
(pd− pN)(T, x, y) = ∞ X r=0 [ep ⊗NH(r) −p ⊗e N HN(r) ](T, x, y)
where we put p ⊗e N HN(r)
(T, x, y) = 0 for hr > T . Summing over r ∈ N in the
identity (p ⊗e NH(r)−ep ⊗N HN(r))(T, x, y) = e p ⊗NH(r−1) ⊗N(H − HN) (T, x, y) + e p ⊗NH(r−1)−p ⊗e NHN(r−1) ⊗N HN (T, x, y) one gets (pd− pN)(T, x, y) =hpd⊗N(H − HN) + pd− pN⊗N HN i (T, x, y). By iterative application of the last identity we obtain
(pd− pN)(T, x, y) = ∞ X r=0 h pd⊗N(H − HN) i ⊗N HN(r)(T, x, y).
We get from (4.10) that for all t ∈ {(ti)i∈[[1,N ]]}:
pd⊗N(H − HN) (T, x, y) = − M−1X k=1 hk (k + 1)! pd⊗N e Φ∗−Φe∗ k+1 e p (T, x, y) −h M M ! Z 1 0 (1 − τ )M pd⊗N e Φ∗−Φe∗ M+1 e pτ (T, x, y)dτ. Eventually, (pd− pN)(T, x, y) = − M−1X k=1 hk (k + 1)! pd⊗N e Φ∗−Φe∗ k+1 pN (T, x, y) +hMRM,2(T, x, y) , RM,2(T, x, y) = − 1 M ! Z 1 0 (1 − τ )M pd⊗N e Φ∗−Φe∗ M+1 e p∆τ (T, x, y) dτ ∀t ∈ {(ti)i∈[[1,N ]]}, pe∆τ(t, x, y) = ∞ X r=0 e pτ⊗N HN(r)(t, x, y) ,pe ∆ 0 = pN.
This proves the expansion part of the Lemma. The bound follows as in the previous proof from Lemma 43.
We now state Lemma 43 that allows to control the rests appearing in the expansions of Lemmas 41 and 42. Its proof is postponed to appendix A.
Lemma 43 Let q > d + 4. For all multi-indices a, b s.t. |a| + |b| < q − (d + 4), the following inequalities hold:
DyaDxbpd(tk, x, y) + DayDbxpN(tk, x, y) ≤ Ct− |a|+|b| α k p(te k, x, y), k ∈ [[1, n]], (4.11) DayDbxp(t, x, y) ≤ Ct−|a|+|b|α p(t, x, y), 0 < t ≤ T.e
Also, ∃C := C(T ) s.t. for all (x, y, z) ∈ (Rd)3, s ∈]0, T ], p(s, x, z) ≤ Cpey(s, x, z), pd(s, z, y) ≤ Cpey(s, z, y), |(Φ −Φe∗)kpd(s, z, y)| ≤ Cs−|k|αpey(s, z, y), t Φ −Φe∗k p(s, x, z) ≤ Cs− |k| αpey(s, x, z). (4.12)
5 Extensions and conclusion
A careful examination of the proofs in the Appendices shows that the absolute conti-nuity of λ w.r.t. to the Lebesgue measure of Sd−1can be removed in (A-1) provided the function ζ(t, x, y) := 1 (2π)d Z Rd ( Z Sd−1 |hp, si|αλf (x)(ds)) exp −t Z Sd−1 |hp, si|αλf (y)(ds) × exp(−ihp, xi)dp has bounded derivatives w.r.t. x up to order q (see Appendix B and the statement of Theorem 3.1 in [Kol00]). Also up to a standard perturbative argument, similar controls on the density can be obtained when we consider (1.1) driven by (Zs+ Ps)s≥0
where (Ps)s≥0 is a compound Poisson process with L´evy measure νP(dz) = f (z)dz
and |f (z)| ≤ 1+|z|Cd+β, β > 0, see Theorem 4.1 in [Kol00]. In that case our main results
remain valid up to a modification of the remainder. Indeed, it is the smallest exponent (or equivalently the largest tail) that leads the asymptotic behavior of p(t, x, y) when |x − y| is large. Thus ρα,M(T, y − x) has to be replaced by ρmin(α,β),M(T, y − x) in
Theorem 21. Eventually, good controls have been obtained on p for stable-like processes, i.e. when the stability index in the generator Φψ(x) in (2.2) can depend on the spatial position x, i.e. α turns to α(x) ∈ [α, α] strictly included in (0, 2] (see Section 5 in [Kol00]). Anyhow the processes associated to those generators cannot be approximated by a usual Euler scheme and the previous analysis breaks down. The approximation of such processes will concern further research.
A Proof of the controls on the derivatives of the densities (Lemma 43) To conclude the proof it remains to prove Lemma 43. The first step is to get bounds on partial derivatives of the transition densities ep and p. The following estimates generalize the ones obtained in [Kol00], Propositions 2.1-2.3.
Lemma A1 Let q > d + 4. There exists a constant C > 1 such that the following estimates
hold uniformly for α in any compact subset of the interval (0, 2) and for all 0 < t ≤ T, x, y, z ∈
Rdand |a| < q − (d + 4) |Dazpey(t, x, z)| ≤ C t|a|/αpe y(t, x, z), (A.1) |Da zpey(t, x, z)| ≤ C |z − B(y)t − x||a|pe y(t, x, z). (A.2)
Remark A1 Equation (A.1) extends to the stable case what is widely known in the Gaussian
framework. Namely, each derivation of the density in space remains homogeneous to a stable density up to a multiplicative additional singularity of order t−1/α.
Proof. From now on we assume w.l.o.g. that d ≥ 3, the cases d ∈ {1, 2} can be addressed more
directly. To proceed with the computations, we need to specify a useful change of coordinates. Namely, for a given direction ζ ∈ Rd\{0} introduce for p ∈ Rd the spherical coordinates
(ρ, ϑ, ϕ2, · · · , ϕd−1), ρ = |p| with first coordinate or main axis directed along ζ, that is
p1 = ρ cos ϑ, p2= ρ sin ϑ cos ϕ2, p3= ρ sin ϑ sin ϕ2cos ϕ3, ...
pd−1= ρ sin ϑ sin ϕ2... sin ϕd−2cos ϕd−1,
pd= ρ sin ϑ sin ϕ2... sin ϕd−2sin ϕd−1, (A.3)
ϑ ∈ [0, π] , ϕi ∈ [0, π] , i ∈ [[2, d − 2]], ϕd−1∈ [0, 2π]. Consider then the coordinates (v, τ, φ)
where τ = cos ϑ and v = ρ |ζ|, with v ∈ R+, τ ∈ [−1, 1], φ = (ϕ
2, · · · , ϕd−1) ∈ [0, π]d−3×
[0, 2π]. In the following we write p = p(v, τ, φ) for the previous r.h.s. in (A.3) written in these new coordinates that is
p1= |ζ|−1vτ, p2= |ζ|−1v(1 − τ2)1/2cos ϕ2,
p3= |ζ|−1v(1 − τ2)1/2sin ϕ2cos ϕ3, ...
pd−1= |ζ|−1v(1 − τ2)1/2sin ϕ2... sin ϕd−2cos ϕd−1,
pd= |ζ|−1v(1 − τ2)1/2sin ϕ2... sin ϕd−2sin ϕd−1, (A.4)
and ¯p(τ, φ) = p(|ζ|, τ, φ).
Without loss of generality we suppose B(y) = 0. The first step consists in differentiating w.r.t z the inverse Fourier transform for epy(t, x, z)
e py(t, x, z) = 1 (2π)d Z Rdexp −t Z Sd−1|hp, si| αλ f (y)(ds)
exp (−i hp, z − xi) dp. (A.5) For z = x, (2.3) and standard computations directly give estimate (A.1). Thus, in the following we also assume z 6= x and use the previous spherical coordinates (v, τ, φ) derived from (A.3) setting ζ = z − x as the main axis. We obtain:
Dazpey(t, x, z) = 1 (2π)d|z − x||a|+d Z∞ 0 dv v|a|+d−1× Z 1 −1 dτ Z [0,π]d−3×[0,2π]dφΨ(v, τ, |a|) exp −t v α |z − x|α Z Sd−1|hp, si| αλ f (y)(ds) × τa1 1 − τ2|a|−a1+d−32 h(φ, a) , (A.6) where p = p/ |p|, a = (a1, ..., ad) ∈ Ndand
Ψ(v, τ, |a|) = (−1)|a|/2cos(vτ )I|a| even+ (−1)(|a|+1)/2sin(vτ )I|a| odd,
h(φ, a) = {(cos ϕ2)a2(sin ϕ2cos ϕ3)a3× ... × (sin ϕ2... sin ϕd−2cos ϕd−1)ad−1
× (sin ϕ2... sin ϕd−2sin ϕd−1)ad} × V (φ),
V (φ) = (sin ϕ2)d−3(sin ϕ3)d−4× ... × (sin ϕd−3)2sin ϕd−2.
We consider, first the case |z − x| /t1/α≤ C, for a sufficiently small positive constant C. In
this case we expand the trigonometric function Ψ(v, τ, |a|) in (A.6) in power series and change the variable of integration t1/αv
|z−x| to w in each term. This gives for all k ∈ N,
Da zpey(t, x, z) = C|a| t|a|+dα Xk m=0 (−1)m (2m + I|a| odd)!e |a| m |z − x| t1/α
2m+I|a| odd
+R|a|k+1
, C|a|= (−1)
(|a|+I|a| odd)/2
where ∀m ∈ [[1, k]], e|a|m = Z ∞ 0 dw Z 1 −1 dτ Z [0,π]d−3×[0,2π]dφ exp −wα Z Sd−1|hp, si| αλ f (y)(ds)
w|a|+2m+d−I|a| even× τa1+2m+I|a| odd 1 − τ2|a|−a1+d−32 h(φ, a),
|R|a|k+1| ≤ |e|a|k+1| (2(k + 1) + I|a| odd)! |z − x| t1/α
2(k+1)+I|a| odd
.
(A.8) To simplify the notations we omit the dependence of the coefficients of our expansions on the direction ζ = z − x. From (A-1), (A-2) and (2.3) one then derives the following bound:
e|a|m ≤ Ad−2 αC |a|+2m+d+I|a| odd α 1
Γ |a| + 2m + d + I|a| odd α B m +a1+ 1 + I|a| odd 2 , |a| − a1+ d − 1 2 . (A.9)
Here Ad−2denotes the area of the unit sphere Sd−2and B is the β-function. Note that the
modulus of each term in the expansion (A.7) serves as an estimate of the remainder in a finite Taylor expansion. From (A.7) we have
Dzapey(t, x, z) = C|a| t|a|+dα e|a|0 |z − x| t1/α I|a|odd + R|a|1 ! . (A.10)
Recall that we are considering the case |z−x|t1/α ≤ C. By Proposition 3.1 (i) from [Kol00] for
some eC depending on C, eC−1t−d/α ≤ epy(t, x, z) ≤ eCt−d/α. Hence, equations (A.10), (A.9),
(A.8) yield |Dzapey(t, x, z)| ≤ C t|a|α e py(t, x, z) ≤ CC |a| |z − x||a|pe y(t, x, z). (A.11) To estimate Da
zpey(t, x, z) for |z − x| /t1/α ≥ (C)−1 we proceed as in Proposition 2.3 of
[Kol00]. This gives the following representation Da
zpey(t, x, z) = [Dzaepy(t, x, z)]1+[Dazepy(t, x, z)]2 with [Dazpey(t, x, z)]j= 1 (2π)d Z∞ 0 dρρ|a|+d−1 Z1 −1dτ Ψ(ρ|z − x|, τ, |a|) × fj(τ ) Z [0,π]d−3×[0,2π]dφ exp n −tραg λf(τ, φ, y) o h(φ, a), j = 1, 2, (A.12) gλf(τ, φ, y) := Z Sd−1|h¯p(τ, φ), si| αλ f (y)(ds),
using the notations introduced after (A.3). Here f1(τ ) = τa1 1 − τ2 |a|−a1+d−3 2 χ(τ ), f 2(τ ) = τa1 1 − τ2 |a|−a1+d−3 2 (1 − χ(τ ))
where χ(τ ) is a C∞even truncation function R → [0, 1] that equals 1 for |τ | ≤ 1 − 2ε, and 0 for
|τ | ≥ 1 − ε for some ε ∈ (0,12). Because of the symmetry in τ , it is easy to see that the integral
in (A.12) is non-zero only if a1 and |a| are both even or odd. Expanding the exponential at
order 2 in (A.12) and making the change of variables ρ |z − x| = v we get [Dzapey(t, x, z)]1= C|a| |z − x||a|+d 2 X m=0 1 m!b |a| m t |z − x|α m , (A.13)
where C|a|is defined in (A.7) and for m ∈ [[0, 1]], b|a|m = (−1)m
Z ∞
0
Fm|a|(v)v|a|+mα+d−1dv,
Fm|a|(v) = [I|a| evenRe − I|a| oddIm]
Z∞ −∞exp(−ivτ )ϕ m(τ )dτ , ϕm(τ ) = f1(τ ) Z [0,π]d−3×[0,2π]g m λf(τ, φ, y)h(φ, a)dφ. and b|a|2 = 2 Z 1 0 (1 − δ) Z ∞ 0 F2,δ|a|(v)v|a|+2α+d−1dvdδ, F2,δ|a|(v) = [I|a| evenRe − I|a| oddIm]
Z ∞ −∞exp(−ivτ )ϕ2,δ (τ )dτ , ϕ2,δ(τ ) = f1(τ ) Z [0,π]d−3×[0,2π]g 2 λf(τ, φ, y) exp −δt v |z − x| α gλf(τ, φ, y) ×h(φ, a)dφ.
To extend the integration to R in the definition of (Fm|a|(v))m∈[[0,1]], F2,δ|a|(v), we simply use
that the functions (ϕm)m∈[[0,1]], ϕ2,δ have compact support in τ . However, to check that
the coefficients (b|a|m)m∈[[0,2]] are well defined, we have to equilibrate at infinity the term in
(v|a|+mα+d−1)
m∈[[0,2]]. This can be done computing iterated integration by parts in τ in
the definition of (Fm|a|(v))m∈[[0,1]], F2,δ|a|(v). Namely, if ϕm(τ ), m = 0, 1, and ϕ2,δ(τ ) are Cq
functions of τ with compact support and q > |a|+4+d > |a|+2α+d, performing q integrations by parts w.r.t. τ one derives that the coefficients (b|a|m)m∈[[0,2]] are well defined. Let us now
check that assumption (A-1) implies that ϕm(τ ), m = 0, 1, and ϕ2,δ(τ ) are Cq functions of τ
with compact support. Indeed, for the unit vectors p(τ + △τ, φ) and p(τ, φ), from elementary algebra there exists an orthogonal matrix A := A(∆τ ) s.t. p(τ +△τ, φ) = Ap(τ, φ). Hence, if λf (x)(ds) = Θ(x, s)ds where Θ has the previous smoothness one can show
lim ∆τ →0 gλf(τ + ∆τ, φ, x) − gλf(τ, φ, x) ∆τ = lim ∆τ →0 R Sd−1{|hp(τ, φ), A∗si|α− |hp(τ, φ), si|α}λf (x)(ds) ∆τ = Z Sd−1|hp(τ, φ), si| α lim ∆τ →0 [Θ(x, As) − Θ(x, s)] ∆τ ds = Z Sd−1|hp(τ, φ), si| αΘ′ s(x, s)β(τ, φ, s)ds,
where β(τ, φ, s) is C∞function in τ uniformly bounded in (τ, φ, s) in our region. The process
can then be iterated other q − 1 times.
Thus all coefficients (b|a|m)m∈[[0,2]] are well defined.
Next, analogously to Proposition 2.3 in [Kol00] (where the case |a| = 0 was considered) and with the same rotations of the integration contours for α ∈ (0, 1], α ∈ (1, 2), we obtain for all k ∈ N [Da zpey(t, x, z)]2= C|a| |z − x||a|+d Xk m=0 1 m!c |a| m t |z − x|α m + R|a|2,k+1 , (A.14) c|a|m = 2[I|a| evenRe − I|a| oddIm][
Z1 1−2ε dτ Z [0,π]d−3×[0,2π]dφh(φ, a)(−gλf(τ, φ)) m × exp(−iπαm 2 )(−i)
|a|+dτ−(αm+d+|a|)Γ (αm + d + |a|) f 2(τ )],
and |R|a|2,k+1| ≤ |c|a|k+1| (k+1)! t |x−z|α k+1
. Note that the coefficients c|a|m are also well defined
because τ does not approach zero (recall that 1 − χ(τ ) 6= 0 ⇔ |τ | > 1 − 2ε). Precisely |c|a|m| ≤ 2Ad−2Cm2 (1 − 2ε)−αm+d+|a|Γ (αm + d + |a|).
Now the sum of expansions (A.13) and (A.14) gives the expansion for Da
zpey(t, x, z). Note
that by construction, the first coefficient b|a|0 + c|a|0 does not depend on the spectral measure λf (y)(·) and it vanishes when the spectral measure is uniform (that is C1= C2= 1 in (2.3)).
This can be shown by means of representations involving Bessel and Whittaker functions and the same rotations of the integration contours as in Proposition 2.2 of [Kol00], see Appendix C for details. Thus, for all k ∈ N∗, we get a representation
Da zepy(t, x, z) = C|a| |z − x||a|+d Xk m=1 1 m!d |a| m t |z − x|α m + R|a|k+1 , (A.15)
where d|a|m = b|a|m + c|a|m with bm= 0 for m ≥ 3 and |R|a|k+1| ≤ |d
|a| k+1| (k+1)! t |x−z|α k+1 . Now, from Proposition 3.1 (ii) in [Kol00] d0
1> 0. Equation (A.15) yields
Da zpey(t, x, z) = C|a|d01t |z − x||a|+d+α d|a|1 d0 1 + eR|a|2 ! , | eR|a|2 | ≤|d |a| 2 | 2d0 1 t |x − z|α, e py(t, x, z) = C0 |z − x|d d0 1t |z − x|α+ R 0 2 ≥ C0d 0 1t 2 |z − x|d+α for sufficiently small C. Hence, we have
|Da zpey(t, x, z)| ≤ CC|a| |z − x||a| d0 1t |z − x|d+α ≤ C |z − x||a|ep y(t, x, z) ≤ CC |a| t|a|/αpe y(t, x, z), (A.16)
recalling that |z−x|t1/α ≤ C for the last inequality. W.l.o.g. we can assume C < 1. It remains to consider the case |x − z|/t1/α∈]C, C−1
[:= I(C). It follows from (A.6) that |z − x|dpey(t, x, z)
and |z −x|d+|a|Da
zpey(t, x, z) are continuous functions of |x−z|/t1/α. Since the stable density is
also strictly positive, we deduce that there exists eC s.t. on I(C), |Da
zepy(t, x, z)| ≤|z−x|Ce|a|+d ≤
C
|z−x||a|pey(t, x, z) ≤
CC|a|
t|a|/αepy(t, x, z) which concludes the proof.
Lemma A2 Let q > d + 4. There exists a constant C > 1 s.t. the following estimates hold
uniformly for α in any compact subset of the interval (0, 2) and for all 0 < t ≤ T, x, y, v ∈ Rd
and |a| + |b| < q − (d + 4): DyaDbxH(t, x, y) ≤ C t|a|+|b|α e p(t, x, y) 1 +min(1, |y − x| t , (A.17) DxbH(t, x, x + v) ≤ C ep(t, x, x + v) 1 +min(1, |v| t , (A.18) DyaDbxp(t, x, y)e ≤ C
|y − B(y)t − x||a|+|b|ep(t, x, y). (A.19)
Proof. Inequalities (A.17) and (A.18) follow from the representation
H(t, x, y) = hB(x) − B(y), ∇xp(t, x, y)i +e 1 (2π)d Z Rd|p| αZ Sd−1|hp, si| α × (λf (y)(ds) − λf (x)(ds)) exp −t |p|α Z Sd−1|hp, si| αλ f (y)(ds) ×
exp {−ihp, y − B(y)t − xi} dp (A.20) analogously to the proof of Proposition 2.3 in [Kol00], see also Appendix B where (A.17) is proved for |a| = |b| = 0. Inequality (A.18) contains in (3.23’) p.748 of that reference. Inequality (A.19) can be derived following the proof of Lemma A1.
The proof of Lemma 43 can then be achieved from Lemmas A1 and A2 adapting the arguments in Appendix B concerning the control in terms of the frozen density for the ”formal” series appearing in (3.4). See also the proof of Theorem 2.3 in [KM02] or Theorem 3.1 in [Kol00].
B Control of the parametrix series of the density
For the sake of completeness we provide in this section a complete proof of the control for the r.h.s of (3.4) under our standing Assumptions (A-1)-(A-3).
We first sum up in Proposition B1 the various estimates needed to control the convergence of (3.4) following the proof of Theorem 3.1 in [Kol00], namely Proposition 3.1 and its corollary, Lemma 3.1 and Propositions 3.2-3.3 of that reference. These estimates can also be directly derived from the computations of Appendix C.
Proposition B1 For all K sufficiently large, there exists C > 0 s.t. the following estimates
hold uniformly for α in any compact subset of (0, 2), for all x, y, z ∈ Rdand for all t ∈ (0, T ].
C−1t−d/α ≤ epy(t, x, z) ≤ Ct−d/α, |x − z| ≤ Kt1/α, C−1t |x − z|d+α ≤ ep y (t, x, z) ≤ Ct |x − z|d+α, |x − z| ≥ Kt 1/α, e pz(t, x, z) ≤ C epy(t, x, z).
Also, there exists C > 0 s.t. ∀(t, x, y) ∈ [0, T ] × (Rd)2,
Z
dz min(1, |z|)epy(t, 0, z) ≤ Ctω, ω := min(1, 1/α).
For all s ∈ (0, t) Z dz epz(t − s, x, z) min(1, |y − z|)s−1epy(s, z, y) ≤ C t−1min(1, |y − x|) + sω−1pey(t, x, y), Z dz min(1, |z − x|)epz(t − s, x, z) × min(1, |y − z|)s−1pey(s, z, y) ≤ Csω−1pey(t, x, y), Z dz epz(t − s, x, z)epy(s, z, y) ≤ C epy(t, x, y). Introduce now for a given bounded measure η on Sd−1the function
ϕη(t, z, λf (y)) := 1 (2π)d Z Rddp|p| α Z Sd−1|h¯p, si| αη(ds) × exp −t|p|α Z Sd−1|h¯p, si| αλ f (y)(ds) exp(−ihp, zi). With this notation and (A.20) we get
Under our standing assumptions, the mean value theorem yields |ϕλf (y)− ϕλf (x))(t, y − x −
B(y)t, λf (y))| ≤ min(1, |y − x|)ϕηx,y(t, y − x − B(y)t, λf (y)), where ηx,yis a bounded measure.
Now Proposition 2.5 in [Kol00] states that for a bounded measure η, ϕη(t, z, λf (y)) ≤ Ct−1pey(t, 0, z).
From Lemmas A1, A2 and the above controls one deduces |H(t, x, y)| ≤ C epy(t, x, y) 1 +
t−1min(1, |x − y|):= Cv(t, x, y) (which actually gives (A.17) for a = b = 0).
Introduce now β ◦ ψ(t, s, x, y) = Z
Rdβ(t − s, x, z)ψ(s, z, y)dz, i.e. ◦ is the spatial part of the
convolution operator ⊗, and set ev(t, x, y) := tv(t, x, y). From Propositon B1 one derives e
p ◦ v(t, s, x, y) ≤ C(v(t, x, y) + sω−1ep(t, x, y)) e
v ◦ v(t, s, x, y) ≤ C(ev(t, x, y) + (sω−1+ (t − s)ω)ep(t, x, y)). Recalling |H(t, x, y)| ≤ Cv(t, x, y), integrating the above inequalities one gets:
|ep ⊗ H|(t, x, y) ≤ C(ev(t, x, y) + tωp(t, x, y)), |ee p ⊗ H ⊗ H|(t, x, y) ≤ C2tω(ep + ev)(t, x, y).
An induction yields, for all k ∈ N∗:
|ep ⊗ H(2k)|(t, x, y) ≤C 2ktkω (k!)2 (ep + ev)(t, x, y), |ep ⊗ H(2k+1)|(t, x, y) ≤ C 2k+1tkω k!(k + Iα∈(0,1])! (tv(t, x, y) + tωp(t, x, y)Ie α>1),
and the the required control, i.e. p(t, x, y) ≤ C ep(t, x, y). The controls on the derivatives can be proved in a similar way, up to suitable rearrangements of the variable of integration, see p.747 and 748 in [Kol00]. Also the whole proof can be carried out for pd, pN.
Remark B1 To conclude, note that by arguments similar to those used to prove Proposition
B1, one gets
|H ⊗ H(t, x, y)| ≤ Ctω−1p(t, x, y),e
which turns to be a useful estimate to derive (4.8) following the above proof.
C Additional computations concerning the derivatives of the density In this section we prove that b|a|0 + c|a|0 = 0, justifying that the first index in (A.15) is one.
Odd dimensions d
From the definitions in the proof of Lemma A1 , it is enough to show
I|a| evenRe − I|a| oddIm Z ∞ 0 Z Rexp (−ivτ ) f 1(τ )dτ v|a|+d−1dv +2 Z 1 1−2ε
(iτ )−(d+|a|)Γ (d + |a|) f2(τ )dτ
= 0. (C.1) Note that since d is odd, if |a| is odd i−(d+|a|) = (−1)d+|a|2 and if |a| is even i−(d+|a|) =
i−1(−1)d−1+|a|2 . Hence, the contribution of the second term in (C.1) vanishes and the condition
writes I
|a| evenRe − I|a| oddIm
Z∞ 0 Z Rexp (−ivτ ) f 1(τ )dτ v|a|+d−1dv = 0. (C.2) Denote G1(v) =RRexp (−ivτ ) f1(τ )dτ . Remind that |a| and a1have the same parity, see proof
a) For even |a|, a1, G1is even and belongs to a Schwartz space of functions. Since d is odd, by
inverse Fourier transform, Equation (C.2) reduces to Z∞ 0 G1(v)v|a|+d−1dv =1 2 Z R G1(v)v|a|+d−1dv =(−i) |a|+d−1(2π)d 2 f (|a|+d−1) 1 (0) = 0.
The equality f1(|a|+d−1)(0) = 0 follows from the Leibniz differentiation rule for the product τa1× (1 − τ2)|a|−a1+d−32 and the identity |a| + d − 1 = a1+ (|a| − a1+ d − 3) + 2]. Thus
(C.2) holds in this case.
b) Analogously, for odd |a|, a1, -Im(G1(v)) is odd and belongs to a Schwartz space of functions.
The function (−ImG1(v)v|a|+d−1) is even. Thus
Z∞ 0 (−ImG 1(v))v|a|+d−1dv =(−i) |a|+d 2 f (|a|+d−1) 1 (0) = 0
for the same previous reasons and equation (C.2) holds in this case as well.
Even dimensions d
We assume in this section that the spectral measure is uniform with C1= C2= 1 in (A-1).
For |a| and a1= 2m even, equation (A.6) can be rewritten as
Da zpey(t, x, z) = (−1)|a|/2Aa d−2 (2π)d|z − x||a|+d m X j=0 (−1)2m−jCj m (C.3) × Z∞ 0 dvv|a|+d−1exp −t v α |z − x|α Z1 −1(1 − τ 2)Nj−1/2cos(vτ )dτ where Aa d−2= R
[0,π]d−3×[0,2π]h(φ, a)dφ and Nj=|a|−a1+d−2+2j2 , j ∈ [[0, m]]. Now recalling
the definition of the Bessel function Jn(z) := (z/2)
n
Γ (n+12)√π
R1
−1(1 − t2)n−1/2cos(zt)dt which is
well defined for n > 1/2 on C\(−∞, 0), we get Da zpey(t, x, z) = (−1)|a|/2Aa d−2 (2π)d|z − x||a|+d m X j=0 (−1)2m−jCj m2NjΓ (Nj+1 2) √ π × Z ∞ 0 v|a|+a1+d−2j2 exp −t v α |z − x|α JNj(v)dv = (−1) |a|/2Aa d−2 (2π)d|z − x||a|+d m X j=0 (−1)2m−jCmjΓ (Nj+ 1 2)2 Nj+1/2 (C.4) ×Re Z ∞ 0 exp −t v α |z − x|α exp 1 2Nj+ 1 4 πi W0,Nj(2iv)v N′jdv, where W0,n(z) = exp(−z/2)Γ (n+1 2) R∞ 0 [t(1 + t/z)]n−1/2e−tdt, n > 1/2, z ∈ C\(−∞, 0), is the
Whit-taker function and for z > 0, Jn(z) = 2Re 1 √ 2πzexp( 1 2(n + 1 2)πi)W0,n(2iz)
(relation (2.10) from [Kol00]) and N′
j = |a|+a1+d−2j−12 , j ∈ [[0, m]]. For α ∈ (0, 1], from
Cauchy’s theorem we can change the integration path in (C.4) to the negative imaginary half line. Setting then v = −iξ we obtain
Dzaepy(t, x, z) = (−1)|a|/2Aa d−2 (2π)d|z − x||a|+d m X j=0 (−1)2m−jCmjΓ (Nj+1 2)2 Nj+1/2 ×(−1)j−mRe −i Z ∞ 0 exp −t ξ α |z − x|αexp −iπα 2 W0,Nj(2ξ)ξ Nj′dξ .
Recalling the definition of W0,Nj, we conclude expanding the exponential in power series that
the first term is 0.
For α ∈ (1, 2), using the same arguments we can rotate the initial contour through the angle −π/(2α). Setting then η = ei2απ v we get
Da zepy(t, x, z) = (−1)|a|/2Aa d−2 (2π)d|z − x||a|+d m X j=0 (−1)2m−jCj mΓ (Nj+1 2)2 Nj+1/2 ×Re Z∞ 0 exp it η α |z − x|α+ 1 2Nj+ 1 4 πi − iπ 2α(N ′ j+ 1) ×W0,Nj(2η exp iπ(α − 1) 2α )ηNj′dη.
Taylor’s formula for exp(it|z−x|ηαα) yields for the first term, ∀j ∈ [[0, m]],
Ij α:= Re exp 1 2Nj+ 1 4 πi − iπ 2α(N ′ j+ 1) Z ∞ 0 W0,Nj(2η exp iπ(α − 1) 2α )ηNj′ dη .
At last, we rotate the contour through the angle −π(α−1)2α . Setting ξ = η expiπ(α−1)2α we obtain Iαj = Re n −i(−1)j−mR∞ 0 W0,Nj(2ξ)ξ N′ jdξo= 0.
For |a| and a1= 2m + 1 odd,
Da zepy(t, x, z) = (−1)(|a|+1)/2Aa d−2 (2π)d|z − x||a|+d m X j=0 (−1)2m−jCj m × Z ∞ 0 dvv|a|+d−1exp −t v α |z − x|α Z 1 −1(1 − τ 2)Nj−1/2τ sin(vτ )dτ = (−1) (|a|+1)/2Aa d−2 (2π)d|z − x||a|+d m X j=0 (−1)2m+1−jCj m Z ∞ 0 dvv|a|+d−2exp −t v α |z − x|α × Z 1 −1(1 − τ 2)Nj−1/2τ d(cos(vτ )) = (−1) (|a|+1)/2Aa d−2 (2π)d|z − x||a|+d m X j=0 (−1)2m−jCjm Z ∞ 0 dvv|a|+d−2exp −t v α |z − x|α × Z 1 −1cos(vτ ) × h (1 − 2Nj)τ2(1 − τ2)Nj−3/2+ (1 − τ2)Nj−1/2 i dτ. The above integrals have the same form as in (C.3) and can be estimated similarly.
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